Type II error of a one-sided proportion test

Andy claims $20\%$ of people love a certain app; Aaron believes the true share is $15\%$. Aaron polls $100$ random people and uses a one-sided large-sample z-test of $H_0:p=0.2$ against $H_a:p=0.15$ at level $0.05$ (critical value $z=-1.645$). What is the probability $\beta$ of a type II error under Aaron's alternative, to three decimals?

$0.671$. The test rejects when $\hat p<0.2-1.645\times0.04=0.1342$. If $p=0.15$, $\hat p$ has standard deviation $\sqrt{0.15\cdot0.85/100}\approx0.0357$, so $\beta=\mathbb{P}(\hat p\ge0.1342)=\mathbb{P}(Z\ge-0.443)\approx0.671$.